Firstly, there shouldn't be much twist in a properly laid anchor chain anyway. Secondly, under any kind of tension a swivel simply won't swivel anyway. And thirdly, it's a recognised weak point - especially with a sideways pull.
I'd like to make some comments on these statements.
Firstly, the twist. I agree that in general the worst that could happen between boat and anchor is a couple of changes of tide and/or wind direction, which will not normally be a problem. However, there is a far more significant source of twist. When a non-symmetrical anchor, which means just about anything other than a Fisherman, is hauled aboard with a modern, fast windlass, it spins quite significantly. In the murky waters of the UK this is not obvious (although the results can be) but in the Med it is seen very easily. The first problem that will occur is that the chain will twist. Some windlasses will allow the chain to rotate over the gypsy, with the result that it becomes increasingly twisted in the chain locker. Eventually the time will come when the chain cannot be lowered as it has jammed itself into a tight bundle. Other windlasses will not allow jumping the gypsy. Instead the chain will jam between gypsy and housing. A swivel does not completely overcome the problem but it helps a lot.
Secondly. It's not important that it swivels under load. It's when the anchor is being hauled, under relatively low load, that it does, going some way to preventing the first problem.
The third point is very misleading. I tested a large number of shackles and swivels for Yachting Monthly. The better swivels were about 1.5 times the strength of chain. Some were poor but I identified the design fault that caused this. The best shackles were almost as strong but the worst, mostly galvanised ones, were very poor, some less than half the strength of chain of the equivalent size.
The lateral loading issue is very simply dealt with and was the subject of another YM article. Inserting a couple of links of chain between the swivel and the anchor completely avoids the lateral loading but provides all the benefits of the swivel.